In this lesson,
we will look at several combinations.
A combination is an arrangement when order does not matter.
A combination is for grouping with no repetition.
In contrast, a permutation is an arrangement
when order does matter.
A permutation is an ordered list with no repetition.
For the first example, a committee has 14 members,
a subcommittee of five members must be chosen.
In how many ways can five members be chosen
for a subcommittee from 14 members?
In this case, we are forming groups of five
from the 14 members where the order does not matter.
And therefore, this is an example of a combination.
The number of five members subcommittees
from the 14 members is equal to 14 choose five.
This is a common notation for a combination.
And so is this notation.
In either case, n is equal to 14 and r is equal to five.
Where n choose r is equal to n factorial
divided by the product of n minus r factorial
and r factorial,
which means 14 choose five is equal to 14 factorial
divided by the product of 14 minus five factorial
and five factorial.
Simplifying this gives us 14 factorial
divided by the product of nine factorial and five factorial.
Let's evaluate this on the calculator two ways.
We'll evaluate this quotient using factorials.
And we'll also evaluate the combination
using the combination function.
And I'll be using the Desmos scientific calculator.
To enter a factorial,
we can use the exclamation mark from the keyboard,
or click on Functions and find the factorial here.
Going back to the main keyboard,
I will enter a 14 factorial divided by nine factorial,
times five factorial,
which gives us 2002 ways
to form the five member subcommittees.
Let's also evaluate this using the combination function,
click Functions, click nCr,
and then type in 14 comma five, and we get the same result.
So there are 2002 ways to form five member subcommittees
from the 14 members.
Next, Trevor has 12 books on his summer reading list.
He wants to choose three books to read over the summer.
In how many ways can he choose the books?
This is also an example of a combination
because it does not specify an order
in which the books will be read.
Trevor is simply making groups of three books
from the 12 with no repetition.
Which means a number of ways he can choose the books
is 12 choose three.
Where n is equal to 12 and r is equal to three.
12 choose three is equal to 12 factorial
divided by the product of 12 minus three factorial
and three factorial.
This gives us 12 factorial
divided by the product of nine factorial
and three factorial.
Going back to the calculator,
we can enter a 12 factorial
divided by nine factorial times three factorial,
which gives us 220.
Or from the Functions menu, click nCr,
and then enter 12 comma three.
And of course we get the same result.
There are 220 ways
that Trevor can select or choose three books
from his 12 books.
Let's look at two more examples.
Some friends are playing a game
that requires each person to receive three cards
from a 52 card deck.
How many different three-card hands are possible?
Well, the order of the cards does not effect the hand
and the cards are not repeated,
and therefore we answer the question using a combination.
The number of three-card hands is equal to 52 choose three.
52 choose three is equal to 52 factorial
divided by the product of 52 minus three factorial
and three factorial,
which is equal to 52 factorial
divided by the product of 49 factorial and three factorial.
Going to the calculator, using factorials,
we have 52 factorial
divided by the productive 49 factorial and three factorial,
which is equal to 22,100.
Or using the combination function,
of course, we get the same result.
There are 22,100 possible three-card hands.
To the last example.
An organization has 14 employees.
There are eight female employees and six male employees.
How many teams of six can be chosen,
where there are two women and four men?
To answer this question,
we will have to use the accounting principle
as well as combinations.
We will first determine
how many ways we can choose two women from eight women.
Then multiply that by the number of ways
we can choose four men from six men.
And that product will give us the number of six person teams
that can be formed
where each team has two women and four men.
So again, the number of ways
we can choose two women from eight is eight choose two.
And then times the number of ways
we can choose for men from six, which is six choose four.
Eight choose two is equal to eight factorial
divided by the product of eight minus two factorial
and two factorial.
And six choose four is equal to six factorial
divided by the product of six minus four factorial
and four factorial.
Simplifying, we have eight factorial
divided by the product of six factorial and two factorial,
times six factorial divided by the product of two factorial
and four factorial.
Going back to the calculator.
Using factorials, we have eight factorial
divided by six factorial, times two factorial,
right arrow times six factorial
divided by the product of two factorial and four factorial,
which gives us 420.
Or using the combination function, we have
nCr and then we enter eight comma two,
right arrow, times
nCr six comma four, which gives us the same result.
There are 420 possible teams.
Where each team has two women and four men.
I hope you found this helpful.